# Worksheet: Linear Transformation Composition

In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

**Q9: **

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

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What is the matrix ?

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What is the matrix ?

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**Q10: **

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

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Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

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What, therefore, is the slope of the line of reflection?

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If lies in the direction of the line of reflection, what is ?

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By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

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**Q14: **

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

- Aa reflection in the line through the origin at a inclination
- Ba reflection in the line through the origin at a inclination
- Ca reflection in the line through the origin at a inclination
- Da reflection in the line through the origin at a inclination
- Ea reflection in the line through the origin at a inclination

**Q15: **

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

- Aa reflection in the line through the origin at a inclination
- Ba reflection in the line through the origin at a inclination
- Ca reflection in the line through the origin at a inclination
- Da reflection in the line through the origin at a inclination
- Ea reflection in the line through the origin at a inclination

**Q16: **

Describe the geometric effect of the transformation produced by the matrix .

- Aa dilation with center the origin and scale factor 3 followed by a reflection in the line
- Ba dilation with center the origin and scale factor 3 followed by a rotation about the origin
- Ca dilation with center the origin and scale factor 3 followed by a reflection in the line
- Da dilation with center the origin and scale factor 3 followed by a rotation about the origin
- Ea dilation with center the origin and scale factor 3 followed by a rotation about the origin

**Q17: **

Which of the following compositions of transformations is represented by the matrix ?

- Aa dilation with center the origin and scale factor 2 followed by a reflection in the line
- Ba rotation by about the origin followed by a reflection in the line
- Ca dilation with center the origin and scale factor 2 followed by a reflection in the line
- Da dilation with center the origin and scale factor followed by a reflection in the line
- Ea dilation with center the origin and scale factor followed by a reflection in the line

**Q18: **

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector to .

Find the matrix representation of the transformation formed.

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Find the scale factor of the original dilation.

- Ascale factor = 169
- Bscale factor = 13
- Cscale factor = 154
- Dscale factor = 13
- Escale factor =

**Q19: **

The unit square, with vertices , and , is transformed by a rotation and then a dilation. Its image under this combined transformation is , as shown in the diagram.

What are the coordinates of ?

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What is the matrix of the combined transformation?

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